How vague is vague? How informative is informative? Reference analysis for Bayesian meta-analysis.

Abstract

Meta‐analysis provides important insights for evidence‐based medicine by synthesizing evidence from multiple studies which address the same research question. Within the Bayesian framework, meta‐analysis is frequently expressed by a Bayesian normal‐normal hierarchical model (NNHM). Recently, several publications have discussed the choice of the prior distribution for the between‐study heterogeneity in the Bayesian NNHM and used several “vague” priors. However, no approach exists to quantify the informativeness of such priors, and thus, we develop a principled reference analysis framework for the Bayesian NNHM acting at the posterior level. The posterior reference analysis (post‐RA) is based on two posterior benchmarks: one induced by the improper reference prior, which is minimally informative for the data, and the other induced by a highly anticonservative proper prior. This approach applies the Hellinger distance to quantify the informativeness of a heterogeneity prior of interest by comparing the corresponding marginal posteriors with both posterior benchmarks. The post‐RA is implemented in the freely accessible R package ra4bayesmeta and is applied to two medical case studies. Our findings show that anticonservative heterogeneity priors produce platykurtic posteriors compared with the reference posterior, and they produce shorter 95% credible intervals (CrI) and optimistic inference compared with the reference prior. Conservative heterogeneity priors produce leptokurtic posteriors, longer 95% CrI and cautious inference. The novel post‐RA framework could support numerous Bayesian meta‐analyses in many research fields, as it determines how informative a heterogeneity prior is for the actual data as compared with the minimally informative reference prior.